Resources

Vortices and Topological Defects

Order Parameters

Much of physics deals with the competition between order and disorder. Energetic typically drive matter towards an ordered state:
the energy of a collection of sodium atoms is minized if it forms a crystal. On the other hand, energy is not the only
concept playing a role in determining the behavior of a physical system. If you are at finite temperature, then entropy favors a disordered state. Similarly, at zero temperature "quantum fluctuations"
may favor a disordered state. [For example, a classical crystal has every atom localized. A corrolory of the Heisenberg uncertainty principle
is that it is hard to truly localize particles. Consequently you have materials such as Helium, which remain liquid down to arbitrarily low temperatures
(at least at atmospheric pressure).]

Ordered states of matter typically have more degrees of freedom than unordered states: one would not
notice if you uniformly translated all of the atoms in a liquid, but such a move would be important
in a crystal. The object which encodes these degrees of freedom is the "order parameter". Order parameter textures occur when the order parameter varies
slowly through space. A particularly interesting form of textures are those that are topological, meaning that they require the order parameter to
be discontinuous. For example a dislocation in a crystal is topological.